| Name |
Jean-Marc J GINOUX |
| Country |
France |
| Email |
ginoux@univ-tln.fr |
| Co-Author(s) |
LLIBRE, ROSSETTO, LOZI, CHUA |
| Submit Time |
2014-02-27 11:15:53 |
Session |
Special Session 114: Nonstandard Analysis, Quantizations and Singular Perturbations
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| Contents |
| The aim of this work is to propose a new method for proving the existence of "canard solutions" for three and four-dimensional singularly perturbed systems with only one fast variable which does not require a center manifold reduction nor a blow-up technique i.e. a desingularization procedure for folded singularities. Contrary to previous works, this method uses the normalized slow dynamics and not the projection of the desingularized vector field. This method enables to state a unique generic condition for the existence of "canard solutions" for such three and four-dimensional singularly perturbed systems which is based on the stability of folded singularities of the normalized slow dynamics and not of the projection of the desingularized vector field. Application of this method to the famous three and four-dimensional memristor canonical Chua's circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to prove the existence of "canard solutions" in such Memristor Based Chaotic Circuits. Moreover, extension of this method to the case of four-dimensional singularly perturbed systems with two fast and two slow variables will be briefly presented. Then, applications of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model will enable to prove as many previous works the existence of "canard solutions" in such system. |
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